The potential of the e⁺e^--collider experiments to search for the signal of a new pseudoscalar Higgs boson and study its CP properties is analyzed. The new Higgs boson coupling to b quarks and t leptons, which is parameterized in a model-independent way, m_b/v(a + ig₅b), is chosen to be an indicator of its CP nature. We analyzed the e⁺e^-t⁺t^-nn, e⁺e^-bnn and e⁺e^-e⁺e^-b processes, which reveal a high sensitivity to anomalous Higgs boson couplings due to the dominant contributions from vector-boson fusion processes W^*W^* H and Z^*Z^* H in central region. It is shown that the polarization study of final state particles could help to separate the contributions from the scalar and pseudoscalar Higgs boson. For instance, the account for the cascade t-lepton decays in the e⁺e^-t⁺t^-nn process allows one to determine the CP state of the Higgs boson as well as the value and sign of the b parameter. As a result of the processes analysis, we develop the strategy of hunting for a new pseudoscalar Higgs boson signal in future linear e⁺e^--collider experiments, which will allow one to understand the CP nature of the Higgs boson and put bounds on its couplings to fermions.

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